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Applied Mathematics

Principles of Quantum Mechanics and Laws of Wave Optics from One Mathematical Formula

Do Tan Si

Published Jan 1, 2019• 0 citations
Publisher

Abstract

Finding that in the formula of expansion of a function into a series of wave-like functions the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schrödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.

DOI
10.4236/am.2019.1011064

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Finding that in the formula of expansion of a function into a series of wave-like functions the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schrödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.
In 1926, the Austrian physicist Erwin Schrodinger derived an equation of waves from the Hamilton variational principle inspired by the analogy between Mechanics and Optics. This equation explained much of quantum phenomenology was known at that time. Physical laws must be invariant under certain symmetries, represented by the transformations of the mathematical object that define these laws. In particular in quantum mechanics one can ask for the transformations of states and observables. These transformations must fulfill what constitutes the essence of the so-called Wigner's theorem, that is, that the transformed observables must possess the same possible sets of eigenvalues as the old ones and that the transformations of the states must give the same probabilities. The temporal evolution of the states must always be given by unit operators that conserve scalar products.
Synopsis: After drilling down to the basic theory, the process of building up the quantum description from experiment is continued. The reader is introduced to the Schrödinger equation and related concepts. The emphasis is on what experiment tells us about particle properties, and various experiments as well as the Bohr model are used to motivate wave–particle dualism ideas. It is argued that the Fourier transform is the mathematical concept needed to build in this dualism. The Schrödinger equation is deduced from an analogy with the wave equation, and connection is made to the ideas of probability. Finally, the entire formalism is recovered and illuminated by reconnecting to the more formal spin basis ideas of the first chapter.
Abstract This chapter examines some crucial experiments and basic principles of quantum mechanics that have strongly influenced the development of quantum theory in its modern form: starting from Planck’s law of black-body radiation and its derivation using the quantization of energy; Einstein’s photon hypothesis for the puzzle presented by the photoelectric effect and its test via the Millikan experiment; the Compton effect; Bohr’s theses for the structure of the hydrogen atom and the complementarity of waves and particles i.e., the wave-particle duality, culminating in de Broglie’s ideas for matter waves and their confirmation in the Davisson-Germer experiment for electron diffraction from a crystal; Heisenberg’s uncertainty principle for position and momentum as well as for energy an time; the famous double-slit experiment and Born’s probability interpretation, and finally the provocative thought experiment of Schrödinger’s cat
Synopsis: After drilling down to the basic theory, the process of building up the quantum description from experiment is continued. ƒe reader is introduced to the Schrödinger equation and related concepts. ƒe emphasis is on what experiment tells us about particle properties, and various experiments as well as the Bohr model are used to motivate wave/particle dualism ideas. It is argued that the Fourier transform is the mathematical concept needed to build in this dualism. ƒe Schrödinger equation is deduced from an analogy with the wave equation and connection is made to the ideas of probability. Finally, the entire formalism is recovered and illuminated by reconnecting to the more formal spin basis ideas of the first chapter.
A century ago, the energy quantization hypothesis and wave-particle duality have been proposed as the foundations of the quantum theory, but still lacking complete theoretical proof and interpretation. For the first time, the "quantized wave particle duality equation" is expressed by a way of physical and mathematical formulas, which simply compose of three factors: the quantized resonant eigenvalue, interaction or transition time, and sum of translational kinetic energy; and accompanied by a full interpretation of the related physical laws. From now on, they are no longer the premise hypothesis of quantum theory, but the inevitable conclusion. The long-term conflict between quantum theory and classical physics has been resolved, which is actually a self-consistent and coherent theoretical system.
Based on the hypothesis that the position-representation of a physical state <x|α\alphaα> is the Fourier transform of its momentum-representation <P|α\alphaα> and that the time-representation <t|α\alphaα> is the inverse Fourier transform of its energy-representation <E|α\alphaα>, we are able to find the Planck relation E = hv, the de Broglie relation P = hλ\frac{h}{\lambda}λh​. Afterward, utilizing the Dirac delta function δ\deltaδ(x) and the property FTxf(x) = -iℏ\hbarℏ∂\partial∂, pFTxf(x) = -iℏ\hbarℏ∂\partial∂, pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [p^,x^\hat{p}, \hat{x}p^​,x^] = -iℏ\hbarℏI^\hat{I}I^, the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.